Free Existential Completion

• Free existential completion is a construction that freely adjoins existential quantifiers to a primary doctrine, extending its logical expressiveness.
• It is characterized by a universal property ensuring unique factorization of maps from the completion to any existential doctrine.
• The construction preserves key logical operations such as finite meets, the Beck–Chevalley condition, and supports witness extraction and choice principles.

Searching arXiv for recent and foundational papers on free existential completion and related doctrine completions. Free existential completion is the doctrine-theoretic construction that adjoins existential quantification to a doctrine in the freest possible way. In the setting of primary doctrines, it takes a functor $P\colon C^{\mathrm{op}}\to\mathbf{Pos}$ or, in the inf-semilattice setting, $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$, and produces an existential doctrine equipped with left adjoints to reindexing along projections, satisfying Beck–Chevalley and, in the inf-semilattice formulations, Frobenius reciprocity. Its central feature is a universal property: maps from the completion into any existential doctrine are uniquely determined by maps from the original doctrine. This places free existential completion at the intersection of categorical logic, exact and regular completion theory, dialectica-style constructions, and algebraic treatments of choice principles (Trotta et al., 2020).

1. Doctrine-theoretic setting

A primary doctrine is a functor $P\colon C^{\mathrm{op}}\to\mathbf{Pos}$ where $C$ has finite products and each fibre $P(A)$ carries an entailment order in context $A$. In the inf-semilattice formulation used in later work, a primary or elementary doctrine is a functor $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$ where fibres have finite meets and top, and reindexing preserves them (Trotta, 2021).

An existential doctrine is obtained when reindexing along product projections has left adjoints interpreted as existential quantifiers. In the primary-doctrine presentation, for each projection

$\mathrm{pr}_i\colon A_1\times A_2\to A_i$

the reindexing $\mathrm{pr}_i^*$ has a left adjoint $\exists_{\mathrm{pr}_i}\dashv \mathrm{pr}_i^*$, and these adjoints satisfy Beck–Chevalley for pullbacks of projections. In the elementary or inf-semilattice presentations, Frobenius reciprocity is also required: $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$0 These are the usual categorical axioms for existential quantification (Trotta et al., 2020).

The phrase “free existential completion” refers to the universal enlargement of a doctrine by such existential structure. Different papers formulate the construction with slightly different ambient 2-categories and classes of maps. Trotta’s treatment focuses on projections in primary doctrines and identifies a 2-monadic structure on the completion (Trotta, 2021). Generalized existential completion replaces projections by a pullback-stable class $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$1, yielding $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$2-existential doctrines and, for $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$3, the full existential completion (Maietti et al., 2021). A further “pure existential” formulation emphasizes the freely generated $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$4-structure inside elementary doctrines and its relation to regular and exact completions (Maietti et al., 2023).

2. Explicit construction

For a primary doctrine $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$5, the free existential completion $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$6 has the same base category $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$7. The fibre over $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$8 consists of triples $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$9 with $P\colon C^{\mathrm{op}}\to\mathbf{Pos}$0 and $P\colon C^{\mathrm{op}}\to\mathbf{Pos}$1. These are read as predicates $P\colon C^{\mathrm{op}}\to\mathbf{Pos}$2 with an implicit existential quantifier $P\colon C^{\mathrm{op}}\to\mathbf{Pos}$3 in front. The order is generated by

$P\colon C^{\mathrm{op}}\to\mathbf{Pos}$4

iff there exists $P\colon C^{\mathrm{op}}\to\mathbf{Pos}$5 such that

$P\colon C^{\mathrm{op}}\to\mathbf{Pos}$6

in $P\colon C^{\mathrm{op}}\to\mathbf{Pos}$7. Reindexing along $P\colon C^{\mathrm{op}}\to\mathbf{Pos}$8 is given by

$P\colon C^{\mathrm{op}}\to\mathbf{Pos}$9

For a projection $C$0, the existential quantifier is defined by

$C$1

regarding $C$2. This is left adjoint to pullback along $C$3 and satisfies Beck–Chevalley (Trotta et al., 2020).

In the inf-semilattice version $C$4, an element of $C$5 is an equivalence class

$C$6

where $C$7 is a projection. The order is

$C$8

iff there exists $C$9 with $P(A)$0 and $P(A)$1. Fibrewise meets and top are defined by pullbacks of projections and meets in $P(A)$2, while reindexing is induced by pullback. If $P(A)$3 is a projection, then

$P(A)$4

This formulation makes Beck–Chevalley and Frobenius reciprocity directly visible (Trotta, 2021).

A related generalized construction replaces projections by a class $P(A)$5 of morphisms closed under composition, containing identities, and stable under pullback. For a $P(A)$6-conjunctive doctrine $P(A)$7, the fibre of $P(A)$8 over $P(A)$9 consists of pairs $A$0 with $A$1, modulo a preorder defined by factorization and reindexing inequalities. Reindexing is by pullback, finite meets are computed pointwise, and existential quantification along $A$2 is given by post-composition: $A$3 Specializing to $A$4 yields the full existential completion $A$5 (Maietti et al., 2021).

This family of descriptions suggests that free existential completion is not a single presentation but a stable pattern: predicates are enlarged by adjoining a witness-object or witness-map, and existential quantification is realized by composition with the map being quantified over.

3. Universal property and 2-categorical form

The defining feature of free existential completion is its adjoint universal property. For the primary-doctrine construction $A$6, the assignment extends to a 2-functor

$A$7

left adjoint to the inclusion of existential doctrines into all doctrines. Equivalently, for any primary doctrine $A$8 and existential doctrine $A$9, precomposition with the unit $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$0 induces an equivalence

$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$1

natural in $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$2 and $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$3. Thus $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$4 exhibits $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$5 as the free way to add existentials to $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$6 (Trotta et al., 2020).

In the inf-semilattice presentation, there is a 2-adjunction

$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$7

with unit

$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$8

and counit at an existential doctrine $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$9,

$\mathrm{pr}_i\colon A_1\times A_2\to A_i$0

This yields a 2-monad $\mathrm{pr}_i\colon A_1\times A_2\to A_i$1 on primary doctrines. Trotta proves that $\mathrm{pr}_i\colon A_1\times A_2\to A_i$2 is lax-idempotent: any two $\mathrm{pr}_i\colon A_1\times A_2\to A_i$3-algebra structures on the same primary doctrine agree up to a unique isomorphism, and any lax morphism of $\mathrm{pr}_i\colon A_1\times A_2\to A_i$4-algebras is uniquely determined by its underlying 1-cell. Accordingly, the 2-category of strict $\mathrm{pr}_i\colon A_1\times A_2\to A_i$5-algebras is equivalent to the 2-category of existential doctrines (Trotta, 2021).

The same pattern persists in the generalized setting. If $\mathrm{pr}_i\colon A_1\times A_2\to A_i$6 denotes the forgetful 2-functor from $\mathrm{pr}_i\colon A_1\times A_2\to A_i$7-existential doctrines to $\mathrm{pr}_i\colon A_1\times A_2\to A_i$8-conjunctive doctrines, then $\mathrm{pr}_i\colon A_1\times A_2\to A_i$9 has a left 2-adjoint $\mathrm{pr}_i^*$0. Concretely,

$\mathrm{pr}_i^*$1

with unit $\mathrm{pr}_i^*$2 and counit $\mathrm{pr}_i^*$3 (Maietti et al., 2021).

A plausible implication is that the universal property, rather than any specific syntactic presentation, is the invariant content of free existential completion. The explicit constructions differ, but each serves as a left adjoint to forgetting existential structure.

4. Preservation of logical structure

A central technical question is which fibrewise logical operations survive passage to the completion. For the primary-doctrine completion, if each fibre $\mathrm{pr}_i^*$4 has finite meets preserved by reindexing, then each $\mathrm{pr}_i^*$5 has finite meets and reindexing preserves them. The meet is given by

$\mathrm{pr}_i^*$6

Under additional hypotheses involving a distributive base category with stable binary coproducts and suitable left adjoints along coproduct injections, fibres of $\mathrm{pr}_i^*$7 also carry binary joins preserved by reindexing. If the associated $\mathrm{pr}_i^*$8 satisfy Frobenius reciprocity, each fibre $\mathrm{pr}_i^*$9 is a distributive lattice (Trotta et al., 2020).

In the inf-semilattice treatment, the completion is constructed from the outset inside $\exists_{\mathrm{pr}_i}\dashv \mathrm{pr}_i^*$0, so finite meets and top are preserved. Beck–Chevalley and Frobenius reciprocity are built into the doctrine $\exists_{\mathrm{pr}_i}\dashv \mathrm{pr}_i^*$1, making it existential in Lawvere’s sense (Trotta, 2021).

A further preservation theorem concerns elementarity. If $\exists_{\mathrm{pr}_i}\dashv \mathrm{pr}_i^*$2 is elementary with fibrewise equality $\exists_{\mathrm{pr}_i}\dashv \mathrm{pr}_i^*$3, then $\exists_{\mathrm{pr}_i}\dashv \mathrm{pr}_i^*$4 is again elementary. The equality predicate in the completion is

$\exists_{\mathrm{pr}_i}\dashv \mathrm{pr}_i^*$5

The induced left adjoint along the diagonal is compatible with the original equality structure, so elementarity is preserved and “the same equality predicate is carried forward” (Trotta, 2021).

The generalized and pure-existential variants likewise preserve the logical structure that is encoded in their hypotheses. In $\exists_{\mathrm{pr}_i}\dashv \mathrm{pr}_i^*$6, finite meets are computed pointwise and the top element is $\exists_{\mathrm{pr}_i}\dashv \mathrm{pr}_i^*$7, while Beck–Chevalley and Frobenius reciprocity hold for the specified class $\exists_{\mathrm{pr}_i}\dashv \mathrm{pr}_i^*$8 (Maietti et al., 2021). In the pure-existential completion $\exists_{\mathrm{pr}_i}\dashv \mathrm{pr}_i^*$9, expressions are generated from elements of $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$00 by finite meets and existential quantifiers along successive projections, modulo exactly the equations imposed by Beck–Chevalley and Frobenius (Maietti et al., 2023).

These results show that the completion is not merely a formal adjunction of $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$01-symbols. It preserves, and in some formulations reflects, substantial lattice-theoretic and equality-theoretic structure already present in the underlying doctrine.

5. Choice principles and normal forms

Free existential completion also supports internal choice principles. In $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$02, every formula is provably equivalent to a prenex-existential form: $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$03 This normal-form statement says that the completion does not merely permit existential quantification; it presents every predicate as one existential block applied to an underlying predicate from the original doctrine (Trotta et al., 2020).

Under mild hypotheses—namely that each fibre $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$04 has a top element and binary meets—the completion satisfies a Rule of Choice. If

$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$05

in $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$06, then there exists $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$07 in $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$08 such that

$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$09

Thus one may extract a witness function $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$10. The dual universal completion yields a “Counterexample Property,” and when existential and universal completions are combined one obtains the full Skolem-choice schema

$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$11

If $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$12 has exponentials and $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$13 is already universal, then $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$14 remains universal; dually, if $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$15 is existential then $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$16 is still existential (Trotta et al., 2020).

These theorems connect the free construction to Skolemization phenomena. In the subobject-doctrine example, the resulting logic is described as the usual “regular logic with Skolem functions” (Trotta et al., 2020). A plausible implication is that free existential completion algebraizes witness extraction in a way that is intrinsic to the doctrine rather than imposed from an external proof calculus.

6. Variants, examples, and applications

Several canonical doctrines arise as existential completions. If $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$17 has finite limits and $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$18 is the constant-true doctrine $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$19, then $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$20 is the usual subobject doctrine $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$21, while $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$22 is the weak-subobject doctrine $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$23. In particular,

$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$24

This identifies familiar doctrines as universal existential completions of a trivial logical base (Maietti et al., 2021).

For $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$25 and $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$26, the fibre $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$27 is equivalent to the doctrine whose predicates are equivalence classes of formulas of the form “$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$28,” and $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$29 is computed by composition with projection. This example is stated to underlie the classical Gödel-dialectica factors via quantifier completion (Trotta et al., 2020).

The construction is tightly linked to regular and exact completions. For an elementary existential doctrine $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$30, its regular completion $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$31 has as objects pairs $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$32 with $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$33, and morphisms are entire functional relations in $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$34. Its exact completion $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$35 is the “tripos-to-topos” of Joyal–Hyland–Pitts (Maietti et al., 2021). In the full existential setting one has

$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$36

and more generally

$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$37

where $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$38 is the Grothendieck category of $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$39 (Maietti et al., 2021).

Trotta’s monadic account extends this exact-completion machinery from existential doctrines to arbitrary elementary doctrines: for any elementary doctrine $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$40, one first performs $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$41, then applies exact completion to obtain $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$42. The resulting composite is a left biadjoint to the functor sending an exact category to its subobject doctrine (Trotta, 2021). This places free existential completion as the first stage in a general doctrine-to-exact-category pipeline.

The pure-existential variant sharpens this connection. For an elementary pure-existential doctrine $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$43, one can form $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$44 and $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$45. If $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$46 is an elementary subdoctrine, then $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$47 iff the canonical graph functor

$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$48

is an equivalence; under the same hypotheses, the induced functor into $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$49 is also an equivalence (Maietti et al., 2023). This characterizes free pure-existential completion in terms of the regular and exact completions of the category of predicates.

The theory also recovers standard semantic examples. For a pca $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$50, if $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$51 is the realizability tripos on $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$52, then $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$53, and hence

$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$54

recovering the category of assemblies and the realizability topos. For a supercoherent locale $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$55, if $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$56 is the localic tripos and $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$57 its frame of supercompact elements, then $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$58, with regular and exact completions recovering $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$59 (Maietti et al., 2021).

7. Later developments and related results

Subsequent work broadens both the logical ambient and the applications of free existential completion. In a distributive-lattice setting, one can construct the free existential completion of a $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$60-doctrine $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$61 even when $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$62 is only assumed small, with binary products and terminal object added when quantifiers along projections are discussed. The fibre $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$63 is built from finite sets of pairs $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$64 with $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$65, ordered by a condition expressed using finite families of arrows and disjunctions. Joins are unions of finite sets, reindexing acts by pullback on the second coordinate, and the quantifier $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$66 sends $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$67 to $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$68. This yields a $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$69-doctrine, again characterized by a left 2-adjoint to the forgetful 2-functor (Wrigley, 21 Aug 2025).

That distributive-lattice version is used to derive a Herbrand-style theorem for coherent logic. For a universal coherent theory $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$70, the quantifier-free syntactic doctrine $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$71 satisfies

$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$72

the doctrine of depth-one formulas. If

$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$73

is provable, then by the ordering in $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$74 there exist closed terms $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$75 such that

$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$76

This is presented as a weak form of Herbrand’s theorem for coherent logic (Wrigley, 21 Aug 2025).

Another later development concerns arithmetic universes. If $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$77 is the doctrine of predicates of a Skolem theory, then $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$78 is a regular category in which all regular epis split, and

$P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$79

Hence Joyal’s arithmetic universe $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$80 is exactly the exact completion of the pure-existential completion $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$81. For the initial Skolem theory in the standard category of ZFC-sets, the fibre of $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$82 over $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$83 is the set of recursively enumerable subsets of $P\colon C^{\mathrm{op}}\to\mathbf{InfSL}$84, and its exact completion recovers the initial arithmetic universe (Maietti et al., 2023).

Taken together, these developments show that free existential completion functions as a unifying mechanism. It links subobjects and weak subobjects, realizability and localic triposes, regular and exact completions, Skolem-style witness extraction, coherent Herbrand phenomena, and arithmetic universes, all within a 2-categorical adjoint framework (Maietti et al., 2021).